A bijection between EW tableaux and permutation tableaux Thomas - PowerPoint PPT Presentation

A bijection between EW tableaux and permutation tableaux Thomas Selig joint work with Jason Smith and Einar Steingr´ ımsson SLC 78, Ottrott 28 March, 2017 Thomas Selig EW tableaux and permutation tableaux

Ferrers diagram Definition A Ferrers diagram is a left-aligned collection of cells with a finite number of rows and columns such that the number of cells in each row is weakly decreasing. (a) F (b) F ′ F ′ is the Ferrers diagram F with an extra column on the left-hand side. Thomas Selig EW tableaux and permutation tableaux

EW -tableaux Definition (Ehrenborg, van Willigenburg 04) An EW -tableau (EWT) T is a 0–1 filling of a Ferrers diagram that satisfies the following properties: 1 The top row of T has a 1 in every cell. 2 Every other row has at least one 0. 3 No four cells of T that form the corners of a rectangle have 0s in two diagonally opposite corners and 1s in the other two. The size of a EW -tableau is one less than the sum of its number of rows and number of columns. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 (c) an EWT (d) not an EWT Thomas Selig EW tableaux and permutation tableaux

EWTs and acyclic orientations G ( F ) F

EWTs and acyclic orientations 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 G ( F ) F

EWTs and acyclic orientations 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 G ( F ) F Thomas Selig EW tableaux and permutation tableaux

EWTs and acyclic orientations 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 G ( F ) F (0 = ↑ = | , 1 = ↓ = | ) Thomas Selig EW tableaux and permutation tableaux

EWTs and acyclic orientations 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 G ( F ) F (0 = ↑ = | , 1 = ↓ = | ) EWT ( F ) ↔ { Ac. Or. of G(F) where top-left vertex = unique source } . Thomas Selig EW tableaux and permutation tableaux

Permutation tableaux Definition (Postnikov 06) A permutation tableau (PT) T is a 0–1 filling of a Ferrers diagram, some of whose bottom-most rows may be empty, satisfying the following properties: 1 Every column of T has a 1 in some cell. 2 If a cell has a 1 above it in the same column and a 1 to its left in the same row then it has a 1. The size of a permutation tableau is the sum of its number of rows and number of columns. 0 0 0 1 0 0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 1 0 0 1 1 1 1 1 0 (e) a PT (f) not a PT Thomas Selig EW tableaux and permutation tableaux

The main result Theorem (Ehrenborg, van Willigenburg 04; S., Smith, Steingr´ ımsson ++) Let F be a Ferrers diagram (possibly with some empty rows). Then the number of PTs of shape F and the number of EWTs of shape F ′ are the same. Thomas Selig EW tableaux and permutation tableaux

The main result Theorem (Ehrenborg, van Willigenburg 04; S., Smith, Steingr´ ımsson ++) Let F be a Ferrers diagram (possibly with some empty rows). Then the number of PTs of shape F and the number of EWTs of shape F ′ are the same. Formulation in different but equivalent form by EW (04). Proof is recursive. Thomas Selig EW tableaux and permutation tableaux

The main result Theorem (Ehrenborg, van Willigenburg 04; S., Smith, Steingr´ ımsson ++) Let F be a Ferrers diagram (possibly with some empty rows). Then the number of PTs of shape F and the number of EWTs of shape F ′ are the same. Formulation in different but equivalent form by EW (04). Proof is recursive. A more bijective proof by Josuat-Verg` es (10). Thomas Selig EW tableaux and permutation tableaux

The main result Theorem (Ehrenborg, van Willigenburg 04; S., Smith, Steingr´ ımsson ++) Let F be a Ferrers diagram (possibly with some empty rows). Then the number of PTs of shape F and the number of EWTs of shape F ′ are the same. Formulation in different but equivalent form by EW (04). Proof is recursive. A more bijective proof by Josuat-Verg` es (10). We present a bijection through permutations. Thomas Selig EW tableaux and permutation tableaux

Map of the proof T ′ ∈ EWT ( n ) T ∈ PT ( n ) shape( T ) = F shape( T ′ ) = F ′ Φ Ψ σ ∈ S n τ ∈ S n ?? ?? ˜ σ ∈ S n CS ED ?? Thomas Selig EW tableaux and permutation tableaux

The map Φ from permutation tableaux to permutations 9 8 6 5 3 0 0 0 1 0 1 1 0 1 1 1 1 2 2 3 0 0 1 1 4 4 1 1 6 5 7 7 9 8 10 10 Label the rows and columns of the permutation tableau; Thomas Selig EW tableaux and permutation tableaux

The map Φ from permutation tableaux to permutations 9 8 6 5 3 0 0 0 1 0 1 1 0 1 1 1 1 2 2 3 0 0 1 1 4 4 1 1 6 5 7 7 9 8 10 10 Label the rows and columns of the permutation tableau; For each 1 ≤ i ≤ n , construct the path from i to σ i : enter the row, resp. column, labelled i from the left, resp. top; traverse cells with 0; turn S → E or E → S when cell is 1; σ i is the label of the edge through which the path exits on the South-East border; Thomas Selig EW tableaux and permutation tableaux

The map Φ from permutation tableaux to permutations 9 8 6 5 3 0 0 0 1 0 1 1 0 1 1 1 1 2 2 3 0 0 1 1 4 4 1 1 6 5 7 7 9 8 10 10 Label the rows and columns of the permutation tableau; For each 1 ≤ i ≤ n , construct the path from i to σ i : enter the row, resp. column, labelled i from the left, resp. top; traverse cells with 0; turn S → E or E → S when cell is 1; σ i is the label of the edge through which the path exits on the South-East border; We get a map Φ( T ) := σ 3 7 2 6 1 4 9 5 8 10 σ i 10 . i 1 2 3 4 5 6 7 8 9 Thomas Selig EW tableaux and permutation tableaux

The map Φ from permutation tableaux to permutations 9 8 6 5 3 0 0 0 1 0 1 1 0 1 1 1 1 2 2 3 7 2 6 1 4 9 5 8 10 Φ σ : σ i 3 0 0 1 1 4 4 i 1 2 3 4 5 6 7 8 9 10 1 1 6 5 7 7 9 8 10 10 For σ ∈ S n , w ex ( σ ) := { 1 ≤ i ≤ n ; i ≤ σ i } (weak excedences). Theorem (Steingr´ ımsson, Williams 07) Let F be a Ferrers diagram of size n with row labels RL ( F ), then Φ : PT ( F ) − → { σ ∈ S n ; w ex ( σ ) = RL ( F ) } is a bijection. Thomas Selig EW tableaux and permutation tableaux

The map Φ is a bijection 9 8 6 5 3 0 0 0 1 0 1 1 0 1 1 1 1 2 2 Φ 3 7 2 6 1 4 9 5 8 10 σ : σ i 0 0 1 1 3 4 4 1 2 3 4 5 6 7 8 9 10 i 6 5 1 1 7 7 9 8 10 10 Φ : PT ( F ) − → { σ ∈ S n ; w ex ( σ ) = RL ( F ) } σ : { 1 , · · · , n } → { 1 , · · · , n } is a bijection (can construct σ − 1 ). w ex ( σ ) = RL ( F ). Φ is a bijection. We can construct Φ − 1 . Thomas Selig EW tableaux and permutation tableaux

Progress of the proof T ′ ∈ EWT ( n ) T ∈ PT ( n ) RL ( T ) shape( T ′ ) = F ′ Φ σ ∈ S n τ ∈ S n ?? w ex ( σ ) σ ∈ S n ˜ ?? Thomas Selig EW tableaux and permutation tableaux

Cyclic Shift CS For σ = σ 1 · · · σ n ∈ S n , define ˜ σ := CS ( σ ) = σ 2 · · · σ n σ 1 . Thomas Selig EW tableaux and permutation tableaux

Cyclic Shift CS For σ = σ 1 · · · σ n ∈ S n , define ˜ σ := CS ( σ ) = σ 2 · · · σ n σ 1 . 3 7 2 6 1 4 9 5 8 10 CS : σ i 1 2 3 4 5 6 7 8 9 10 i CS ˜ 7 2 6 1 4 9 5 8 10 3 σ i 1 2 3 4 5 6 7 8 9 10 i Thomas Selig EW tableaux and permutation tableaux

Cyclic Shift CS For σ = σ 1 · · · σ n ∈ S n , define ˜ σ := CS ( σ ) = σ 2 · · · σ n σ 1 . 3 7 2 6 1 4 9 5 8 10 CS : σ i 1 2 3 4 5 6 7 8 9 10 i CS ˜ 7 2 6 1 4 9 5 8 10 3 σ i 1 2 3 4 5 6 7 8 9 10 i For ˜ σ ∈ S n , exc (˜ σ ) := { 1 ≤ i ≤ n ; i < ˜ σ i } ((strong) excedences). Proposition For any 2 ≤ a 1 < · · · < a k , CS : { σ ∈ S n ; w ex ( σ ) = { 1 , a 1 , · · · , a k }} − → { ˜ σ ∈ S n ; exc (˜ σ ) = { a 1 − 1 , · · · , a k − 1 }} is a bijection. Thomas Selig EW tableaux and permutation tableaux

Progress of the proof T ′ ∈ EWT ( n ) T ∈ PT ( n ) RL ( T ) shape( T ′ ) = F ′ Φ σ ∈ S n τ ∈ S n w ex ( σ ) ?? σ ∈ S n ˜ CS exc (˜ σ ) Thomas Selig EW tableaux and permutation tableaux

σ �→ τ The map ED : ˜ Algorithm: inputs ˜ σ , outputs τ . 0. Initialise τ = ∅ . 1. j = min { 1 ≤ i ≤ n ; i �∈ τ } . If j = + ∞ return τ . Else j ′ = j and proceed to 2. σ k = j ′ . τ ← τ ∗ k . If k � = j then j ′ = k and 2. Find k s.t. ˜ repeat 2. Else return to 1. Thomas Selig EW tableaux and permutation tableaux

σ �→ τ The map ED : ˜ Algorithm: inputs ˜ σ , outputs τ . 0. Initialise τ = ∅ . 1. j = min { 1 ≤ i ≤ n ; i �∈ τ } . If j = + ∞ return τ . Else j ′ = j and proceed to 2. σ k = j ′ . τ ← τ ∗ k . If k � = j then j ′ = k and 2. Find k s.t. ˜ repeat 2. Else return to 1. σ : ˜ 7 2 6 1 4 9 5 8 10 3 σ i Example: ˜ i 1 2 3 4 5 6 7 8 9 10 Thomas Selig EW tableaux and permutation tableaux

σ �→ τ The map ED : ˜ Algorithm: inputs ˜ σ , outputs τ . 0. Initialise τ = ∅ . 1. j = min { 1 ≤ i ≤ n ; i �∈ τ } . If j = + ∞ return τ . Else j ′ = j and proceed to 2. σ k = j ′ . τ ← τ ∗ k . If k � = j then j ′ = k and 2. Find k s.t. ˜ repeat 2. Else return to 1. σ : ˜ 7 2 6 1 4 9 5 8 10 3 σ i Example: ˜ i 1 2 3 4 5 6 7 8 9 10 τ = Thomas Selig EW tableaux and permutation tableaux